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Kolmogorovkomplexität [Elektronische Ressource] / vorgelegt von Rupert Hölzl

98 pages
INAUGURALDISSERTATIONzurErlangung der DoktorwürdederNaturwissenschaftlich-MathematischenGesamtfakultätderRuprecht-Karls-Universität HeidelbergVorgelegt vonDiplom-Mathematiker Rupert HölzlausMünchen.Tag der mündlichen Prüfung: 16. Dezember 2010ThemaKolmogorovkomplexitätGutachter: Priv.-Doz. Dr. Wolfgang MerkleProf. Dr. Frank StephanKolmogorov complexityby Rupert HölzlEnglish abstract: This dissertation discusses new results on Kolmogorov com-plexity. Its first part focuses on the study of Kolmogorov complexity without timebounds. Here we deal with the concept of non-monotonic randomness, that israndomness characterized by martingales that bettonically. We will statethe definitions of several different randomness classes and then separate them fromeach other. We also present a a systematic survey of a wide array of traceabilitynotions and characterize them through (auto)complexity notions. Traceabilities area group of notions that express that a set is not far away from being computable.The second part of the document deals with the topic of time bounded Kol-mogorov complexity. First we investigate the difference between two ways ofdescribing a word: the complexity of describing it well enough so that it can bedistinguished from other words; and the complexity of describing it well enough sothat the word can actually be produced from the description.
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INAUGURALDISSERTATION
zur
Erlangung der Doktorwürde
der
Naturwissenschaftlich-Mathematischen
Gesamtfakultät
der
Ruprecht-Karls-Universität Heidelberg
Vorgelegt von
Diplom-Mathematiker Rupert Hölzl
aus
München.
Tag der mündlichen Prüfung: 16. Dezember 2010Thema
Kolmogorovkomplexität
Gutachter: Priv.-Doz. Dr. Wolfgang Merkle
Prof. Dr. Frank StephanKolmogorov complexity
by Rupert HölzlEnglish abstract: This dissertation discusses new results on Kolmogorov com-
plexity. Its first part focuses on the study of Kolmogorov complexity without time
bounds. Here we deal with the concept of non-monotonic randomness, that is
randomness characterized by martingales that bettonically. We will state
the definitions of several different randomness classes and then separate them from
each other. We also present a a systematic survey of a wide array of traceability
notions and characterize them through (auto)complexity notions. Traceabilities are
a group of notions that express that a set is not far away from being computable.
The second part of the document deals with the topic of time bounded Kol-
mogorov complexity. First we investigate the difference between two ways of
describing a word: the complexity of describing it well enough so that it can be
distinguished from other words; and the complexity of describing it well enough so
that the word can actually be produced from the description. While this difference
is unimportant in the case of Kolmogorov complexity without time bounds it plays
an essential role when time bounds are present. Next, we introduce the notion of
computational depth and prove a dichotomy result about it that is reminiscent of
Kummer’s well-known gap theorem. Lastly, we look at the important notion of
Solovay functions. Solovay functions are computable upper bounds of Kolmogorov
complexity that are actually sharp infinitely often. We will use them, first, to charac-
terize Martin-Löf randomness in a certain way and, second, to give a characterization
of being jump-traceable.Deutsche Zusammenfassung: In dieser Dissertation werden neue Ergebnisse über
Kolmogorovkomplexität diskutiert. Ihr erster Teil konzentriert sich auf das Studium
von Kolmogorovkomplexität ohne Zeitschranken. Hier beschäftigen wir uns mit
dem Konzept nicht-monotoner Zufälligkeit, d.h. Zufälligkeit, die von Martingalen
charakterisiert wird, die in nicht-monotoner Reihenfolge wetten dürfen. Wir werden
in diesem Zusammenhang eine Reihe von Zufälligkeitsklassen einführen, und diese
dann von einander separieren. Wir präsentieren außerdem einen systematischen
Überblick über verschiedene Traceability-Begriffe und charakterisieren diese durch
(Auto-)Komplexitätsbegriffe. Traceabilities sind eine Gruppe von Begriffen, die
ausdrücken, dass eine Menge beinahe berechenbar ist.
Der zweite Teil dieses Dokuments beschäftigt sich mit dem Thema zeitbe-
schränkter Kolmogorovkomplexität. Zunächst untersuchen wir den Unterschied
zwischen zwei Arten, ein Wort zu beschreiben: Die Komplexität, es genau genug
zu beschreiben, damit es von anderen Wörter unterschieden werden kann; sowie
die Komplexität, es genau genug zu beschreiben, damit das Wort aus der Beschrei-
bung tatsächlich generiert werden kann. Diese Unterscheidung ist im Falle zeit-
unbeschränkter Kolmogorovkomplexität nicht von Bedeutung; sobald wir jedoch
Zeitschranken einführen, wird sie essentiell. Als nächstes führen wir den Begriff der
Tiefe ein und beweisen ein ihn betreffendes Dichotomieresultat, das in seiner Struk-
tur an Kummers bekanntes Gap-Theorem erinnert. Zu guter Letzt betrachten wir
den wichtigen Begriff der Solovayfunktionen. Hierbei handelt es sich um berechen-
bare obere Schranken der Kolmogorovkomplexität, die unendlich oft scharf sind.
Wir benutzen sie, um in einem gewissen Zusammenhang Martin-Löf-Zufälligkeit zu
charakterisieren, und um eine Charakterisierung von Jump-Traceability anzugeben.Contents
Contents 7
1 Introduction 9
1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Thanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Preliminaries 13
I Kolmogorov complexity without time bounds 17
3 Non-monotonic Randomness 19
3.1 Permutation and injection randomness . . . . . . . . . . . . . . . . . . . . 21
3.2 Randomness notions based on total computable strategies . . . . . . 23
3.3 R notions based on partial com strategies . . . . . . 31
4 Traceability and complexity 39
4.1 Traceability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Autocomplex and complex sets . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Diagonally non-computable sets . . . . . . . . . . . . . . . . . . . . . . . 47
4.4 Equivalences of the almost everywhere notions . . . . . . . . . . . . . 48
4.5 Equivalence of the infinitely often notions . . . . . . . . . . . . . . . . 50
4.6 Computable traces and total machines . . . . . . . . . . . . . . . . . . . 52
4.7 Lower bounds on initial segments complexity . . . . . . . . . . . . . . 54
4.8 Tiny use and autocomplexity . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.9 Time bounded traceability and complexity . . . . . . . . . . . . . . . . 58
II Kolmogorov complexity with time bounds 61
5 Distinction Complexity 63
5.1 Known results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7CONTENTS
5.3 The linearly exponential case . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.4 The polynomial case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.5 Space bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6 Kolmogorov complexity and computational depth 77
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.2 Time bounded Kolmogorov complexity and strong depth . . . . . . 80
7 Time bounded complexity and Solovay functions 85
7.1 Solovay functions and Martin-Löf randomness . . . . . . . . . . . . . . 86
7.2 Solovay functions and jump-traceability . . . . . . . . . . . . . . . . . . . 91
Bibliography 95
8CHAPTER 1
Introduction
Since the 1930s, mathematicians such as Gödel, Church and Turing have considered
the notion of computable (or decidable) sets. That is, subsets of the natural numbers
N that can be described in an effective way using only a finite amount of information.
Trivially, every finite set is computable; but there are also many infinite sets that
can be described in such a way. Being computable then means that the set somehow
exhibits enough internal structure and regularity, that despite its infinite cardinality
a finite amount of information suffices to describe it.
Of course, the set of subsets ofN in uncountable whereas a countable list of all
finite descriptions can be given. So it is obvious that all but countably many subsets
ofN cannot be computable.
So how difficult is it to describe more sets? To investigate this the notion of
Kolmogorov complexity was introduced by R.J. Solomonoff, A.N. Kolmogorov
1and G.J. Chaitin. Assume we want to describe some non-computable set A. If we
look at initial segments A?i, that is, the sets A\f0,..., ig for increasing i, how
much information do we need to describe those?
Of course, we can always describe such an initial segment of length i by giving a
sequence of i values inf0,1g. But for some sets A we can actually use fewer bits than
this trivial bound, namely for those sets that, while not exhibiting enough regularity
to be computable, still exhibit enough structure so that we can economise.
As it turned out, there are many non-computable sets exhibiting such regularity
and Kolmogorov complexity is a useful tool to investigate and describe them.
Sequences that do not exhibit such regularities are called random and have been
the central object of study in algorithmic randomness. Many interesting insights
in this area have resulted from trying to relate various notions of randomness with
various notions of computational power[DH10, LV08, Nie09].
In the last decades, a variant of Kolmogorov complexity came into focus: In
this variant, not all space-saving descriptions of sets A are eligible, but only those
1See[LV08] for a more detailed account of the history of the notion.
91. INTRODUCTION
that can be (in some sense) quickly executed to output A. This is know as time-
bounded Kolmogorov complexity. This notion can act as a liaison between the
investigation of Kolmogorov complexity and that of classical structural complexity
theory. The maybe most important difference between Kolmogorov complexity
with time bound and that without is that Kolmogorov complexity with time bound
is itself a computable function.
1.1 Summary
The purpose of this dissertation is to give some new results on Kolmogorov com-
plexity. It essentially consists of two parts.
Part I, with the exception of a short digression in chapter 4, focuses on the study
of Kolmogorov complexity without time bounds. The first chapter in this part
is chapter 3, which deals with the concept of non-monotonic randomness, that is
randomness characterized by martingales that bettonically. We will state
the definitions of several different randomness classes and then separate them from
each other.
In chapter 4 we present a systematic survey of a wide array of traceability notions
and characterize them through (auto)complexity notions. Traceabilities are a group
of notions that express that a set is not far away from being computable.
Part II deals with the topic of time bounded Kolmogorov complexity. Chapter
5 is concerned with the difference between two ways of describing a word: the
complexity of describing it well enough so that it can be distinguished from other
words; and the complexity of describing it well enough so that the word can actually
be produced from the description. While this difference is unimportant in the case
of Kolmogorov complexity without time bounds it plays an essential role when
time bounds are present.
The next chapter, chapter 6, introduces the notion of computational depth and
proves a dichotomy result about it that is reminiscent of Kummer’s well-known gap
theorem[DH10, Kum96].
The last chapter 7 deals with the important notion of Solovay functions. Solovay
functions are computable upper bounds of Kolmogorov complexity that are actually
sharp infinitely often (up to an additive constant). We will use them, first, to charac-
terize Martin-Löf randomness in a certain way and, second, to give a characterization
of being jump-traceable.
1.2 Publications
The work presented in chapter 3 has been published in the proceedings of the
6th International Conference on Computability and Complexity in Analysis in
Ljubljana in 2009 [BHKM09] and will soon appear in the Journal of Logic and
Computation[BHKM]. The largest part of chapter 4 has been published in the
proceedings of the IFIP Conference on Theoretical Computer Science in Brisbane
10

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